The 25-year-old midfielder played under City boss Joe Royle when he was manager of the Merseyside club.
The former England Under-21 international, who agreed to personal terms on Thursday morning, was tipped for a bright future in the game when Royle brought him into the first-team squad.
However, Grant only made 61 first-team appearances at Everton and has already spent a loan spell Tranmere this season, following an injury lay-off .
Royle is keen to include Grant in his squad when City play Leeds in the FA Cup on January 9. Royle is also reported to be interested in bringing out-of-favour Leeds midfielder Alf Inge Haaland to Maine Road.
Jonny Wootton
